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Sep 20, 2013
09/13

by
Pascal Hubert; Samuel Lelièvre

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We study the congruence problem for subgroups of the modular group that appear as Veech groups of square-tiled surfaces in the minimal stratum of abelian differentials of genus two.

Source: http://arxiv.org/abs/math/0410595v1

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7.0

Sep 23, 2013
09/13

by
Nicolas Bedaride; Pascal Hubert

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We consider the billiard map in the hypercube of $\mathbb{R}^d$. We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3d-3}$ is the order of magnitude of the complexity.

Source: http://arxiv.org/abs/1109.6410v1

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Sep 18, 2013
09/13

by
Pascal Hubert; Samuel Lelièvre

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It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two all translation surfaces in H(2) tiled by a prime number n > 3 of squares fall into exactly two Teichmuller discs, only one of them with elliptic points, and that the...

Source: http://arxiv.org/abs/math/0401056v2

5
5.0

Sep 20, 2013
09/13

by
Pascal Hubert; Thomas A. Schmidt

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Veech groups uniformize Teichm\"uller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end is the limit of directions of inequivalent infinite ends.

Source: http://arxiv.org/abs/math/0410132v1

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Sep 19, 2013
09/13

by
Pascal Hubert; Thomas A. Schmidt

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We show that Y. Cheung's general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates. The saddle connection continued fractions then allow one to recognize certain transcendental directions by...

Source: http://arxiv.org/abs/1010.3475v1

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9.0

Sep 23, 2013
09/13

by
Xavier Bressaud; Pascal Hubert; Alejandro Maass

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In this article we prove that given a self-similar interval exchange transformation T, whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of M. Cobo in [C].

Source: http://arxiv.org/abs/0801.2088v1

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5.0

Sep 23, 2013
09/13

by
Yitwah Cheung; Pascal Hubert; Howard Masur

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We consider billiards in a (1/2)-by-1 rectangle with a barrier midway along a vertical side. Let NE be the set of directions theta such that the flow in direction theta is not ergodic. We show that the Hausdorff dimension of the set NE is either 0 or 1/2, with the latter occurring if and only if the length of the barrier satisfies the condition of P'erez Marco, i.e. the sum of (loglog q_{k+1})/q_k is finite, where q_k is the the denominator of the kth convergent of the length of the barrier.

Source: http://arxiv.org/abs/0810.3956v3

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10.0

Sep 18, 2013
09/13

by
Pascal Hubert; Erwan Lanneau; Martin Moeller

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We prove that the Teichmueller disc stabilized by the Arnoux-Yoccoz pseudo-Anosov diffeomorphism contains at least two closed Teichmueller geodesics. This proves that the corresponding flat surface does not have a cyclic Veech group. In addition, we prove that this Teichmueller disc is dense inside the hyperelliptic locus of the connected component H^odd(2,2). The proof uses Ratner's theorems. Rephrasing our results in terms of quadratic differentials, we show that there exists a holomorphic...

Source: http://arxiv.org/abs/math/0611655v2

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20

Sep 17, 2013
09/13

by
Pascal Hubert; Samuel Lelievre; Serge Troubetzkoy

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We study periodic wind-tree models, unbounded planar billiards with periodically located rectangular obstacles. For a class of rational parameters we show the existence of completely periodic directions, and recurrence; for another class of rational parameters, there are directions in which all trajectories escape, and we prove a rate of escape for almost all directions. These results extend to a dense $G_\delta$ of parameters.

Source: http://arxiv.org/abs/0912.2891v1

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Jul 20, 2013
07/13

by
Yann Bugeaud; Pascal Hubert; Thomas A. Schmidt

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We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Source: http://arxiv.org/abs/1007.2050v1

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Sep 21, 2013
09/13

by
Pascal Hubert; Martin Schmoll; Serge Troubetzkoy

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For a Veech surface (x,\omega), we characterize subspaces of X^n, invariant under the diagonal action of the affine group of X. We prove that non-arithmetic Veech surfaces have only finitely many invariant subspaces of very particular shape (in any dimension). Among other consequences we find copies of (X,\omega) embedded in the moduli-space of translation surfaces. We study illumination problems in (pre-)lattice surfaces. For (X,\omega) prelattice we prove the at most countableness of points...

Source: http://arxiv.org/abs/math/0602394v2

5
5.0

Sep 18, 2013
09/13

by
Pascal Hubert; Erwan Lanneau; Martin Moeller

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In this paper, we investigate the closure of a large class of Teichm\"uller discs in the stratum Q(1,1,1,1) or equivalently, in a GL^+_2(R)-invariant locus L of translation surfaces of genus three. We describe a systematic way to prove that the GL^+_2(R)-orbit closure of a translation surface in L is the whole of L. The strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of Ratner's theorem to unipotent subgroups acting on an...

Source: http://arxiv.org/abs/0707.0738v1

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6.0

Sep 21, 2013
09/13

by
Xavier Bressaud; Alexander I. Bufetov; Pascal Hubert

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Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. We consider the corresponding eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the ergodic sums is bounded for every point in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals...

Source: http://arxiv.org/abs/1106.2666v1

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Jul 20, 2013
07/13

by
Vincent Delecroix; Pascal Hubert; Samuel Lelièvre

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The periodic wind-tree model is an infinite billiard in the plane with identical rectangular scatterers disposed at each integer point. We prove that independently of the size of the scatterers, generically with respect to the angle, the polynomial diffusion rate in this billiard is 2/3.

Source: http://arxiv.org/abs/1107.1810v3

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6.0

Sep 20, 2013
09/13

by
Yitwah Cheung; Pascal Hubert; Howard Masur

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In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction either the flow in that direction is completely periodic or minimal, and yet have minimal but non uniquely ergodic directions.

Source: http://arxiv.org/abs/math/0607179v2

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9.0

Sep 18, 2013
09/13

by
Pascal Hubert; Luca Marchese; Corinna Ulcigrai

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We introduce Lagrange Spectra of closed-invariant loci for the action of SL(2,R) on the moduli space of translation surfaces, generalizing the classical Lagrange Spectrum, and we analyze them with renormalization techniques. A formula for the values in such spectra is established in terms of the Rauzy-Veech induction and it is used to show that any invariant locus has closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are dense. Moreover we show that Lagrange spectra of...

Source: http://arxiv.org/abs/1209.0183v2